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They could give me some suggestions on how to use the data on the derivative, in:

If $f:\mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ class is $C^{1}$ such that $\Vert f^{\prime}(x)v\Vert =\Vert v\Vert$, for all $v \in \mathbb{R}^{m}$. then $f$ is isometry.

thank you very much

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The assumption implies that the map $f$ is locally invertible, and that its local inverses satisfy the same condition. Since locally both $f$ and $f^{-1}$ preserve the length of curves, the map $f$ is a local isometry. Any local isometry in $\mathbb{R}^n$ is given by a composition of a translation and an orthogonal map. From this you can conclude that the map $f$ is globally the composition of a translation and an orthogonal map, i.e., a global isometry.

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many Thanks ... –  helmonio Oct 25 '12 at 4:26

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